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Chapter 7: Problem 61

Simplify. \(\sqrt[3]{-16}\)

Short Answer

Expert verified
The simplified form is \(-2\).

Step by step solution

01

Understand Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, \(\theta\theta\theta=8\) implies \(\theta=\theta\).
02

Identify the Given Number

The given number is \(-16\). We are trying to find the cube root of \(-16\).
03

Write Down the Expression

We need to simplify \(\theta\theta\theta\theta\theta\)
04

Simplify the Cube Root

The cube root of \(-16\) is \(-2\), because \(\theta\theta\theta\theta\theta\)= \2\ , 蟽2\theta\theta\ 8\l\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Cube Roots
Let's talk about cube roots. A cube root of a number is a value that, when multiplied by itself three times, returns the original number.
For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In general, for any number \(n\), the cube root is denoted as \(\theta = \sqrt[3]{n}\).

Cube roots are different from square roots because they involve three factors instead of two. This makes them useful in different mathematical and real-world applications.
Exploring Root Properties
Root properties help us understand how to manipulate and simplify roots.
When dealing with cube roots, an important property is that the cube root of a product equals the product of the cube roots. Mathematically, \(\theta\theta\theta\theta\theta\theta =\ \theta\theta\theta\theta*'\theta\).
For example, \(\sqrt[3]{-8 \times -2}= \sqrt[3]{-8} \times \sqrt[3]{-2}\).

These properties are vital for simplifying more complex roots when combined. They help us break down the problem into simpler parts which are easier to handle.
Working with Negative Numbers
Simplifying cube roots can include handling negative numbers. An interesting property of cube roots is that they can indeed handle negation.
Simply put, \(\sqrt[3]{-n} = -\sqrt[3]{n}\).
Why? Because if a negative number is multiplied by itself three times, the result is also negative. For instance, \(-2 \times -2 \times -2 = -8\).

This property simplifies the cube root of any negative number by placing the negative sign outside. It ensures that our calculations stay consistent and produce correct results.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent positive real numbers. $$ -5 x^{7 / 6}\left(x^{5 / 6}-x^{-1 / 6}\right) $$

Rationalize each denominator. Assume that all radicals represent real numbers and that no denominators are \(0 .\) $$ \frac{3}{\sqrt{x+y}} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{144 x^{3} y^{9}}\)

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[5]{x^{3}} \cdot \sqrt[4]{x} $$

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ -\sqrt{\frac{98 r^{3}}{s^{5}}} $$
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